NASA introduced the idea of female astronauts much earlier than you might realize. After all, the Soviets had launched a female cosmonaut!

In the late 1950s, the United States government contemplated training women as astronauts, and newly released medical test results show that they were just as cool and tough as the men who went to the moon.

“They were all extraordinary women and outstanding pilots and great candidates for what was proposed,” said Donald Kilgore, a doctor who evaluated both male and female space flight candidates at the Lovelace Clinic, a mid-century center of aeromedical research. “They came out better than the men in many categories.”

The times being what they were, the program was scrapped, and US women did not make it into space until 1983. Link

I think part of the choice to use men was probably also because of the high chance of fatal failure. In those early days of space tech, the chance of death was highly likely and the space program got a ton of publicity. The American public, especially back then, was used to the idea of male soldiers doing brave things and being injured or killed in the process. The idea of women going on daring missions and being killed or coming back disfigured was not already an accepted part of the culture. Accidently killing a "soldier" and accidently killing a "girl" would have made for very different headlines and reactions.

It would have been tragic if John Glen was burned up on re-entry to the atmosphere or Neil Armstrong had been marooned to starve on the moon. However, the country would have been more apt to accept the losses of those heros and keep on working on and funding and supporting the space program. The space program was able to continue after fatal accidents like the death of Gus Grissom. If a woman, the sex that most Americans at the time associated overwhelming more with being a loving wife or a nurturing mother or a sweet little daughter rather than a tough soldier, were to be killed in the space program, the public backlash might have been too great to keep the program's support up.

Let's be clear. This was a private venture on the part of Lovelace--NOT NASA. It received neither funding nor endorsement from the agency, which was unaware of the whole thing. While the women involved were subjected to many of the same tests the male astronauts endured, and many passed with superior marks, that's where it stopped. They were led to believe it would lead further, but not by NASA! Whether NASA should have involved women is another issue entirely. But we have to recall that it was an era in which such a notion was alien, to put it mildly. So let's stop cooing over the story and recognize it for what it was: a private venture, strictly done by Lovelace, who withdrew the whole thing when the Navy (where he was going to send the women for more testing) asked NASA for a charge code and the agency said "hunh?", at which point the good Dr. Lovelace quietly called the experiment quits. Start reading beyond the web, people.